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But because particle 2 is initially at rest, this equation becomes
The components of the velocities along the -axis have the form . Because particle 1 initially moves along the -axis, we find .
Conservation of momentum along the -axis gives the following equation:
where and are as shown in [link] .
Along the -axis, the equation for conservation of momentum is
or
But is zero, because particle 1 initially moves along the -axis. Because particle 2 is initially at rest, is also zero. The equation for conservation of momentum along the -axis becomes
The components of the velocities along the -axis have the form .
Thus, conservation of momentum along the -axis gives the following equation:
The equations of conservation of momentum along the -axis and -axis are very useful in analyzing two-dimensional collisions of particles, where one is originally stationary (a common laboratory situation). But two equations can only be used to find two unknowns, and so other data may be necessary when collision experiments are used to explore nature at the subatomic level.
Suppose the following experiment is performed. A 0.250-kg object is slid on a frictionless surface into a dark room, where it strikes an initially stationary object with mass of 0.400 kg . The 0.250-kg object emerges from the room at an angle of with its incoming direction.
The speed of the 0.250-kg object is originally 2.00 m/s and is 1.50 m/s after the collision. Calculate the magnitude and direction of the velocity and of the 0.400-kg object after the collision.
Strategy
Momentum is conserved because the surface is frictionless. The coordinate system shown in [link] is one in which is originally at rest and the initial velocity is parallel to the -axis, so that conservation of momentum along the - and -axes is applicable.
Everything is known in these equations except and , which are precisely the quantities we wish to find. We can find two unknowns because we have two independent equations: the equations describing the conservation of momentum in the - and -directions.
Solution
Solving for and for and taking the ratio yields an equation (in which θ 2 is the only unknown quantity. Applying the identity , we obtain:
Entering known values into the previous equation gives
Thus,
Angles are defined as positive in the counter clockwise direction, so this angle indicates that is scattered to the right in [link] , as expected (this angle is in the fourth quadrant). Either equation for the - or -axis can now be used to solve for , but the latter equation is easiest because it has fewer terms.
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